Morten Hjorth-Jensen
466 lines
29 KiB
HTML
466 lines
29 KiB
HTML
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'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
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('Summary from last week, using gradient descent methods, '
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None,
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('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
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('Same code but now with momentum gradient descent',
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("But noen of these can compete with Newton's method",
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'Descent',
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('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
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<a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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<ul class="dropdown-menu">
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<!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
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<!-- navigation toc: --> <li><a href="#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And with loops</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0024"></a>
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<!-- !split -->
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<h2 id="automatic-differentiation" class="anchor">Automatic differentiation </h2>
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<p><a href="https://en.wikipedia.org/wiki/Automatic_differentiation" target="_self">Automatic differentiation (AD)</a>,
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also called algorithmic
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differentiation or computational differentiation,is a set of
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techniques to numerically evaluate the derivative of a function
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specified by a computer program. AD exploits the fact that every
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computer program, no matter how complicated, executes a sequence of
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elementary arithmetic operations (addition, subtraction,
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multiplication, division, etc.) and elementary functions (exp, log,
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sin, cos, etc.). By applying the chain rule repeatedly to these
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operations, derivatives of arbitrary order can be computed
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automatically, accurately to working precision, and using at most a
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small constant factor more arithmetic operations than the original
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program.
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</p>
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<p>Automatic differentiation is neither:</p>
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<ul>
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<li> Symbolic differentiation, nor</li>
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<li> Numerical differentiation (the method of finite differences).</li>
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</ul>
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<p>Symbolic differentiation can lead to inefficient code and faces the
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difficulty of converting a computer program into a single expression,
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while numerical differentiation can introduce round-off errors in the
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discretization process and cancellation
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</p>
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<p>Python has tools for so-called <b>automatic differentiation</b>.
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Consider the following example
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</p>
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$$
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f(x) = \sin\left(2\pi x + x^2\right)
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$$
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<p>which has the following derivative</p>
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$$
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f'(x) = \cos\left(2\pi x + x^2\right)\left(2\pi + 2x\right)
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$$
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<p>Using <b>autograd</b> we have</p>
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<div class="cell border-box-sizing code_cell rendered">
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<div class="input">
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<div class="inner_cell">
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<div class="input_area">
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<div class="highlight" style="background: #f8f8f8">
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<pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
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<span style="color: #408080; font-style: italic"># To do elementwise differentiation:</span>
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> elementwise_grad <span style="color: #008000; font-weight: bold">as</span> egrad
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<span style="color: #408080; font-style: italic"># To plot:</span>
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
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<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">f</span>(x):
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<span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>sin(<span style="color: #666666">2*</span>np<span style="color: #666666">.</span>pi<span style="color: #666666">*</span>x <span style="color: #666666">+</span> x<span style="color: #666666">**2</span>)
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<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">f_grad_analytic</span>(x):
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<span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>cos(<span style="color: #666666">2*</span>np<span style="color: #666666">.</span>pi<span style="color: #666666">*</span>x <span style="color: #666666">+</span> x<span style="color: #666666">**2</span>)<span style="color: #666666">*</span>(<span style="color: #666666">2*</span>np<span style="color: #666666">.</span>pi <span style="color: #666666">+</span> <span style="color: #666666">2*</span>x)
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<span style="color: #408080; font-style: italic"># Do the comparison:</span>
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x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">0</span>,<span style="color: #666666">1</span>,<span style="color: #666666">1000</span>)
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f_grad <span style="color: #666666">=</span> egrad(f)
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computed <span style="color: #666666">=</span> f_grad(x)
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analytic <span style="color: #666666">=</span> f_grad_analytic(x)
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plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">'Derivative computed from Autograd compared with the analytical derivative'</span>)
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plt<span style="color: #666666">.</span>plot(x,computed,label<span style="color: #666666">=</span><span style="color: #BA2121">'autograd'</span>)
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plt<span style="color: #666666">.</span>plot(x,analytic,label<span style="color: #666666">=</span><span style="color: #BA2121">'analytic'</span>)
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plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">'x'</span>)
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plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">'y'</span>)
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plt<span style="color: #666666">.</span>legend()
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plt<span style="color: #666666">.</span>show()
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<span style="color: #008000">print</span>(<span style="color: #BA2121">"The max absolute difference is: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">"</span><span style="color: #666666">%</span>(np<span style="color: #666666">.</span>max(np<span style="color: #666666">.</span>abs(computed <span style="color: #666666">-</span> analytic))))
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</pre>
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